Nuclear Physics BI70 [FSI] (1980) 16-31 © North-Holland Publishing Company

A P H Y S I C A L P I C T U R E F O R T H E P H A S E T R A N S I T I O N S IN Z N

S Y M M E T R I C M O D E L S

Martin B. EINHORN 1 and Robert SAVIT 2

Physics Department, University of Michigan, Ann Arbor, Michigan 48109, USA

Eliezer RABINOVICI

Physics Department, Hebrew University, Jerusalem, Israel

Received 14 February 1980

We show how the phase transitions in Z N symmetric spin and gauge theories can be understood as being caused by condensations of topological excitations. For the two-dimensional Z N periodic ganssian and vector Potts models, there are two phase transitions, the first caused by a condensation of strings (domain boundaries) and the second by an unbinding of vortices. The relationship between our picture and the double Coulomb gas representation of Kadanoff is discussed. Using our representation, we also explain the correspondence between these models and the recent theory of two-dimensional melting of Halperin and Nelson. Finally, we describe generalizations of our picture to higher dimensions and to gauge theories.

1. Introduction

Theor ies with a Z N s y m m e t r y are of interest to a large class of physicists . In

e l emen ta ry par t ic le physics, the impor t ance of the center (Z3) of the SU(3) color

gauge group has been emphas i zed b y ' t Hoof t [1] and Po lyakov [2], who have

a rgued tha t quark conf inemen t m a y be charac te r ized as t r ial i ty conf inement , so

that s tudies of the phases of the Z 3 gauge theories m a y lead to i m p o r t a n t insights.

Moreover , it m a y be that the phase s t ructure of the Z 3 gauge theory in four

d imens ions is re la ted to that of the Z 3 spin m o d e l in two d imens ions [3, 8-10] . The

Z N gauge theory also arises na tu ra l ly in a cer ta in l imit of the abe l i an Higgs mode l

with charge N [4].

In condensed ma t t e r physics Z N symmet r ic theories a p p e a r na tu ra l ly in the

p r o b l e m of two-d imens iona l mel t ing [5]. A crys ta l in two d imens ions has discrete

ro ta t iona l invar iance in the p lane of the crystal . F o r a square la t t ice this s y m m e t r y

is Z 4, while for a t r iangular la t t ice the s y m m e t r y is Z 6. Par t of the p r o b l e m of

mel t ing is assoc ia ted with the res to ra t ion of full ro ta t iona l s y m m e t r y which can be

i Work supported by the Department of Energy contract number EY-76-C-02-1112. 2 Work supported by the National Science Foundation grant number PH78-08426.

16

M . B . Einhorn et al. / Phase transitions 17

broken to a Z 4 o r Z 6 symmetry at low enough temperatures. In the problem of

two-dimensional helium films [6], Z N symmetric interactions are of interest, since the substrate on which the film is placed may have a crystalline structure which gives rise to Z N symmetric forces acting on the helium.

For all these examples, the central issue is to understand the phase structure of the various Z N symmetric theories. In many abelian and non-abelian theories, there are coherent excitations which have topological significance. Frequently, the phase transitions of these theories can be understood as being caused by a condensation of these topological excitations [7]. Familiar examples (reviewed briefly below) include the Ising and x - y models. In this paper, we describe the phase structure of Z N symmetric spin and gauge theories in a similar way.

Recently, a number of papers have appeared discussing the likely phase proper-

ties of these theories [8-10]. In particular, convincing evidence has been amassed, based both on qualitative and quantitative considerations, that for N~> 5 there are three phases. Our considerations were motivated by this work and are consistent with their conclusions; however, our point of view is rather different. Although we shall describe the topological excitations in detail below, at this point we would like to emphasize that they are not entirely the same as the charges of the electric and magnetic Coulomb gas representation discussed by Kadanoff [11]. Rather, all the topological objects in our picture have a "magnet ic" character and have a simple, immediate relationship to the original degrees of freedom of the model.

To illustrate our approach we will, in sect. 2, focus on a two-dimensional globally Z N invariant theory which one may call the "Z N periodic gaussian model"

(PGM)*. This theory has the same topological excitations and is believed to have the same phase structure as the Z s vector Potts model** but is somewhat more tractable. Indeed, in sect. 2 we shall sometimes regard it as a kind of simplified representation of the vector Potts model. On the other hand, the Z ~ P G M is of physical interest in its own right, and it is the simplest non-trivial Z N theory which has all of the features we wish to stress. In sect. 3 we shall describe generalizations of our picture to higher dimensions and to gauge theories.

Before discussing our picture of the Z N theories, it will be useful to review briefly how the critical properties of the two-dimensional Ising and x - y models can be understood as condensations of topological excitations. In the case of the x - y

model [13], the topological excitations are vortex points in two dimensions. An analysis of the hamiltonian for this model (using either duality transformations or other methods) indicates that the vortices at low temperatures occur in vor tex- antivortex pairs bound together by a logarithmic potential. However, the entropy for such a pair is also logarithmic in their separation, and at a certain temperature, To, the free energy is minimized when the vortex and the antivortex are separated

* This form of representation was originally introduced for the continuous planar spin model by Berezinski [12a] and further developed by Villain [12b].

** This model also goes under the name "clock model".

18 M.B . Einhorn et al. / Phase transitions

by an arbitrarily large distance. Thus, above T c the vortices behave like a plasma of free charges. This condensation into a plasma phase causes the x-y spin-spin correlation function, F, to behave like e -a r for large r, r being the separation between the spins, whereas for T < T c F ~ r -n, where 7/is a function of T.

It is possible to understand the phase transition of the d - - 2 Ising model in terms of topological excitations also. In this case the topological excitations are just the domain boundaries between islands of aligned spins. These domain boundaries form closed loops on the links of the dual lattice. (We always have in mind square or cubic lattices: the dual lattice is obtained by shifting the original lattice by half a lattice spacing in each direction.) This is easily seen as follows. Each link of the dual lattice crosses one and only one link of the original lattice. For each nearest neighbor pair of Ising spins which point in opposite directions color the corre- sponding dual lattice link red. It is then easy to see that any configuration of Ising spins on the original lattice gives rise to a configuration of closed red loops on the

dual lattice. Moreover, up to a global flipping of all the spins, the correspondence

is one to one. Now these domain boundaries have a finite energy per unit length. At low

temperatures the dominant configurations are small closed loops occurring with a fairly small density (i.e., at low temperatures most spins point in the same direction). However, a closed loop of a given total length has an entropy propor- tional to its length (modulo logarithms) [14]. Thus there is a competit ion in the free energy of a domain boundary between the energy and the entropy, both of which are proportional to L. At a certain temperature the minimum of the free energy shifts from L = 0 to L = oo (in this crude approximation), and it becomes favorable

to create relatively many closed loops of arbitrary size. But this is just a description of the disordered phase of the Ising model-- i .e . , many misaligned nearest neighbor pairs. Thus the phase transition can be thought of as being produced by a condensation of the domain boundaries which are the topological excitations of

this model*. In the rest of this paper we shall proceed to understand the phase transitions of

Z N symmetric theories in an analogous way. Because Z N is intermediate between U(1) and Z 2, we will find that the topological excitations have characteristics of those found in U(1) theories as well as in Ising-like systems. This gives rise to rather interesting conclusions about the kinds of configurations that are likely to dominate Z N symmetric theories in various phases.

The outline of the rest of the paper is as follows. In sect. 2 we discuss the Z N P G M in two dimensions. Subsect. 2.1 is devoted to a brief review of the phase properties of this model. In subsect. 2.2 we describe the topological excitations of

* Using this picture, a crude approximation to the string entropy consists of supposing that the number of configurations is given by a non-backtracking, but otherwise unrestricted, random walk. Balancing the entropy and energy for a string of length L, one calculates a critical temperature

fl =½ ln3~0 .55 , which is rather close to the exact value of fl =½ ln(l + V 2 ) ~ 0 . 4 4 .

M . B . Einhorn et al. / Phase transitions 19

the model and write the partition function directly in terms of these topological

variables. In subsect. 2.3 we show how the phase transitions of the model can be

understood as condensations of the topological excitations of the model. In sect. 3 we present several comments which include the application of our picture to Z n

spin theories in higher dimensions and to gauge theories, the relationship of our

picture to the double Coulomb gas picture of Kadanoff [11] and the close relationship of our work to theory of two-dimensional melting presented by Halperin and Nelson [5].

2. The two-dimensional Z N spin model

2.1. REVIEW OF PHASE PROPERTIES

The two-dimensional Z N P G M and the two-dimensional vector Potts model are believed to have the same phase structure. In this section we will describe these models and will review what this phase structure is expected to be. For simplicity

we will generally work in the context of the Z N PGM, but one should bear in mind also the vector Potts model.

In the Z s vector Potts model one represents the two-dimensional spin vector by a complex phase, e i(2~r/N)q with 0 < q < N - l, at each site of a square lattice. The

partition function is

Z = ~ exp fl cos - ~ - A , q i , (2.1) (qffi0}

where i is a two-dimensional vector labelling the lattice sites, and A ~ q i = - - q i - q l - f ,

(for simplicity, we drop the vector notation on i). The sum in the exponent is over all nearest neighbor pairs on the lattice, and fl is the inverse temperature. This theory is invariant under the global rotation qi---> qi + a , where a is an integer, and also under the "local" transformation q i - - > q i - N A i where A i is an integer. This local symmetry just expresses the fact that the hamiltonian in (2.1) is periodic.

A more mathematically tractable model having essentially the same structure is • the Z N P G M defined by the partition function

Z = ~ e x p - ½ f l ~ - - ~ - A , q i + 2 ~ r l ~ , ; i . (2.2) (qffi0} (tffi - oo}

Once again there is an integer valued variable, q/, on each site of the lattice as well as an integer valued variable l~; i on each link of the lattice. The sum in the exponent is over all lattice links. After summing over all l~; i from - oo to + oo, it is clear that the logarithm of the argument of the sum over (q} will be invariant under qi-->qi + a , as well as qi-->qi - N A r T h e second, local invariance is precisely

20 M.B. Einhorn et al. / Phase transitions

the result of summing over the l~, r Thus (2.2) considered as a function of the q;'s has the same symmetry as (2.1). (As a function of qi and 1~; i, (2.2) has something like a local gauge symmetry with l~; i playing the role of a vector potential.) In the limit that N--> ~ , (the x-y model limit) (2.2) approximates (2.1) at sufficiently low temperatures (up to an overall field-independent factor).

For general N the model (2.2) (and (2.1)) is thought to have two phase transitions as a function of /3 [8-10]. At low temperatures (large 13) the system is in a ferromagnetic state. Define s i = e i (2~r/N)qi and G ( r ) = (SiSj~ , r = I / - - j l . Then, in

this phase G(r)-->const. ( ~ 0) as r---~ oo, and ( s i~ v ~ O. At a certain value of r , ill, there is a phase transition and the system passes into a phase characterized by ( s i ) = 0 and G ( r ) ~ r -~ as r---> ~ with , / a function of ft. Thus, in this phase the ferromagnetic order disappears, but the system is not totally disordered since G ( r )

falls to zero only like a power and not like an exponential. The power-law behavior of G ( r ) signals the appearance of a massless excitation in this phase, in close analogy with the low-temperature phase of the x-y model. As the temperature is increased still further one encounters another phase transition at /3--/32. For /3 </32 the system is in a true disordered state with ( s i ) = 0, and G ( r ) ~ e -~r for large r. For N < N c (N c ~ 5) [8-10] this scenario breaks down and there is only one phase transition connecting the low- and high-temperature phases described above.

The middle (massless) phase disappears. Because the Z~v PGM is self-dual, the two phase transitions are related to each

other by/31fl2 = N2/4"/r2" As N-~oo we should recover the U(1) PGM, which is expected to have the same phase structure as the x-y model. From these and other considerations, we anticipate that flE--->/3v, the critical point of the U(1) periodic gaussian model, and/31 ~ oo as N--~ oo. That is, the low-temperature, ferromagnetic phase of the Z~v model disappears, and we are left with a massless phase extending from T = 0 to T v ------fl~-I (in which U(1) vortices of equal and opposite vorticity are bound in pairs) followed by a massive disordered phase (in which U(1) vortices are unbound) for T > T v. The simplest dependence of /3t and fiE on N which realizes this behavior is fll ~ NE/(4~'2f lv) and fiE ~/3~ for large enough N.

2.2. TOPOLOGICAL EXCITATIONS

We turn now to a discussion of the excitations of the Z N symmetric theories with topological significance. For our first intuitive arguments it is most convenient to couch the discussion in terms of the complex spins of the vector Potts model, (2.1). Precisely analogous excitations are also present in the model (2.2) as we shall see

explicitly below. First we note that since Z~ is a discrete symmetry we can have closed domain

boundaries separating islands of aligned spins in analogy with the Ising model. Imagine, for example, a nearest neighbor pair connected by a lattice link in direction/~ of Z N spins which are rotated by a relative angle A~q (e.g., by 2or~N). We may associate a piece of domain boundaryj~ = e~,A~,q with the dual lattice link

M.B . Einhorn et al. / Phase transitions 21

which crosses the link joining these two spins. As in the two-dimensional Ising

model it is clear that we will be able to generate closed domain boundaries on the dual lattice using this prescription, since A~j~ = 0. But these domain boundaries differ in two important respects f rom those of the Ising model. First, the bound- aries can have different strengths (or flux). If the orientations of two nearest neighbors differ by an angle whose magnitude is 2~rh/N, where h is an integer and

0 < 2 h / N < 1, then the domain boundary separating them has strength h. One can also consider a boundary of strength h to be a superposition of h boundaries of strength one. Second, for N ) 3 the domain boundaries are orientable; that is, they have an arrow on them. Consider for instance a region, R, of spins oriented at an angle 2 ~r/N surrounded by spins pointing at an angle zero. R is thus circumscribed by a domain boundary of unit flux. If we now rotate all spins in R so that they point at an angle - 2 ~r/N we will still have R circumscribed by a boundary of unit flux, but it will have the opposite sense. Thus we must associate an arrow with a domain boundary. Such a phenomenon does not occur in the Ising (Z2) case.

Since for N ) 3 the spins are complex phases, we might suppose that the Zlv theory also contains vortex-like excitations, in addition to the strings (domain boundaries) described above. Suppose, in analogy to the x-y model, that we wish to place a vortex with vorticity one at some site on the dual lattice. We then require that ~A~(2~r/N)q i = 27r for any closed loop surrounding the vortex, in the nota- tion of (2.1). What is a minimum energy spin configuration which satisfies this requirement? Recall that for the x-y model a minimum energy vortex configuration divides the required 27r rotation more or less equally among all nearest neighbor pairs as we transverse a circle of any radius with the vortex at the center. Thus spins a distance r away from the vortex center are rota ted from their neighbors by an angle ~ 1/ r . The fact that the spins can be rotated from their neighbors by an infinitesimal amount gives rise to an energy for a single x-y model vortex which diverges logarithmically with the size of the system.

Return now to the Z N case. Consider an area with diameter of order N lattice spacings surrounding the location of our Z N vortex. Within this region the Z N spins can divide the required 2~r rotation equally among themselves as we traverse a circle surrounding the vortex. This minimizes the energy density here, and thus in this region it is difficult to tell the difference between a Z N vortex and an x-y model vortex, in that the typical minimum energy spin configurations look the same this close to the vortex center. Outside of this region, however, the Z N spins can no longer equally divide the rotation by 2q r, and the best we can do is illustrated in fig. 1. Our vortex looks like a bicycle wheel with a hub of diameter O ( N ) lattice spacings. The spokes are domain boundaries separating approximately wedge- shaped regions of aligned spins. As we cross a domain boundary the spins change their orientation by 2~/N, so that traversing the N spokes the spins will have rotated by 27r. (Notice that the arrows on the domain boundaries indicate that the spin rotation always has the same sense.)

22 M.B. Einhorn et al. / Phase transitions

Fig. 1. A single Z N vortex for the case N = 6.

Since the domain boundaries have finite energy per unit length, the energy of a single Z s vortex diverges linearly with the size of the system. Thus single vortex configurations should be suppressed, particularly at low temperatures. However, we can also create vortex-antivortex pairs with a finite energy as in the x-y model. Such a pair is shown in fig. 2. The strings joining the vortices are, again, the domain boundaries which have a finite energy per unit length. This gives rise to a linear potential between the vortices. This should be compared with the logarithmic

potential between x-y model vortices. Figs. 1 and 2 suggest that the domain boundaries can somehow be considered as

currents, the divergences of which are associated with the vortices. We will now show that the strings (domain boundaries) and vortices form a complete set of variables in terms of which the partition function can be written.

For the Z N PGM, this is very simple because the vortices and domain bound- aries are easily expressed in terms of the original degrees of freedom. First we

define a vectorj~; i

JtL; i ~ e ~ v ( Arqi + NI,,; i) . (2.3a)

The Jr; ; are naturally associated with links of the dual lattice. Note that

(2~r )2 4 ¢ ; 2 / . 2 4 ~ 2 1 . 2 --~-A qi+2,n.l~; i = ~ e ~ , j p ; i ) = - ~ - - ~ j ~ ; i ) . (2.3b)

Consequently, the partition function (2.2) becomes simply

Z = ~ ~ e x p [ 2~'2fl ~ (j.; i)2] (2.4) (q) {t} [ N2 < > _

where the sum in the exponent is now over dual lattice links.

v

Fig. 2. A Z N vortex-ant ivortex pair for the case N = 7.

M.B. Einhorn et al. / Phase transitions 23

The vectorj~; i is the topological current. The current itself represents the domain boundaries between Z N spins of different orientation and its divergences

Apj#; i = N%,A~I~; i (2.5)

are the vortices (fig. 1). (The extra factor of N should cause no worry: it is clear that when A~j~; i = N the Z N spins rotate through 2rr around the dual lattice thus producing a vortex of vorticity one.) Indeed, if we define 0~ = (2~r/N)q i, then in the limit N ~ oo, 0 i becomes a continuous spin angle, and eq. (2.2) becomes the partition function for the periodic gaussian approximation of the x-y model whose vortex field is given by %,A~l~; i.

We were able to make an immediate association between the original degrees of freedom of the ZN PGM and its topological currents because the Z N periodicity of the PGM is explicitly represented by a field, l~; i. In other Z N theories of interest, for example the vector Potts model, this is not the case, and it is not possible to express the theory in terms of its topological excitations by direct substitution. However, for such theories one can use a more general procedure involving duality transformations to write the theory in terms of its topological excitations. We now illustrate this method for the Z N PGM. applying the standard duality transforma- tion [7] to the model (2.2) we arrive at the dual form

N--1 ~ [ Z = ~ ~ exp - -

ricO m~;i~--oo

N z [2~rA~ri z] (2.6)

where r i, mu; i a r e integer fields defined on the vertices and links respectively of the dual lattice. Note that the Z N PGM is self-dual [15]. We may write this as a Fourier series

E l i ,r N-I 2qr2fl -2 exp i2'n'Y~ --~-Jv;i Z = ~ exp N2 E J v ; i

j , ; i = - m ri=O ( > ( > (2.7)

Thus, Jp;i is a vector field associated with links of the dual lattice, and A j~ is conjugate to the dual spins r. Performing the sum on r i yields the constraint

A A; i = 0 mod N . (2.8)

The general solution of the constraint equation is given precisely by eq. (2.3a). [Indeed, substituting (2.3a) into (2.7) gives back (2.2).]

In view of our intuitive discussion initiating this section, we see that the current J~;i itself represents the topological excitations, i.e., domain boundaries between regions having different orientations of Z N spins, and its non-trivial divergences are precisely the vortices.

24 M.B. Einhorn et al. / Phase transitions

2.3. PHASE TRANSITIONS AND TOPOLOGICAL EXCITATIONS

We will now describe the phase properties of the model, (2.2) or (2.4) in terms of the topological current and its divergence. Since the Z N symmetry is discrete, we expect that at very low temperatures (fl>> 1) the system will be in a ferromagnetic state with (ei(2*r/N)q~ =t= O. In terms of the current, J~;i this phase will have a

relatively small density of relatively small closed domain boundaries, as well as a small density of tightly bound vor tex-ant ivor tex pairs. Recall that the vortices are

bound by a linear potential. (The current, j~; ~, has a finite energy per unit length so that the energy between two divergences connected by strings must grow linearly

with the separation.) Note also that the strings binding the vortices will want to proceed from the vortex to the antivortex in almost the shortest way possible. On the other hand, the flux will prefer to be spread across a transverse region of order N lattice spacings, since the exponent in (2.4) depends on the (flux) 2. It is thus energetically favorable to have no more than one unit of flux on any lattice link. From these considerations it is clear that vortex pairs separated by a distance L take at least N times as much energy to be produced as a closed domain boundary of perimeter L. So for low temperatures closed strings should dominate in size and

number over vor tex-ant ivor tex pairs. As the temperature is increased the size and density of closed domain boundaries

and vortex-ant ivor tex pairs will increase. We now ask whether either one or both of these kinds of excitations are able to condense into a plasma-like phase. Insofar as the intervortex potential is linear the vortices will be prevented from unbinding

since the entropy of a vortex pair is only logarithmic (we will come back to this below). However, we can anticipate a condensation of the closed domain boundaries. Let us restrict our attention to domain boundaries of unit flux. Higher flux boundaries will be present but are energetically disfavored, at least at low temperatures. If we ignore the orientability of the domain boundaries, then we have a situation very much like that of the two-dimensional Ising model discussed earlier. The entropy of the domain boundaries is given by a kind of modified random walk, and so we anticipate a phase transition at some temperature T~ = fl~-~. If we approximate the number of allowed configurations of a domain boundary as a non-backtracking random walk then we can approximate the free

energy of such a string by

F ( L ) ~ 2vr2fl L N 2 -- L In 3 . (2.9)

For fl > fll " ~ N 2 In 3/2 'n "2, F(L) has its minimum at L = 0, while for fl < i l l the minimum of F ( L ) is at L = ~ . This crude argument indicates a condensation of the domain boundaries at a temperature fl~ ~ N 2 In 3 / 2 ~ "2.

We do not expect that this simple estimate of fll will be very accurate. First, in (2.9) we have neglected modifications to the random walk which more closely

M. B. Einhorn et al. / Phase transitions 25

describes the actual situation. For instance, we should not allow walks which tread on a previously used link. (More precisely, such repeated walks have a higher energy.) Restrictions like these will reduce the entropy for long strings which means that the real value of T I will be higher than our estimate. A second point, related to the one we have just mentioned, is that we have completely neglected the orienta- bility of the strings. Including the orientability adds a length independent factor to the entropy of a closed string (i.e., the number of possible configurations is multiplied by 2 independent of the length of the string) and so at the level of the argument leading to (2.9) does not change the estimate of ill. However, in the real system the strings do overlap and interact and the energy associated with their overlaps does depend on their relative orientation and so could affect the value of

ill. In a moment we will return to the discussion of the numerical value of 131, but

now let us ask how the system behaves for fl < ill- For fl < 131 the free energy (2.9) favors configurations with a relatively high density of strings of arbitrary length. This means that it is very easy for the q; to change by one unit as we move from lattice site to lattice site. Now, we are interested in the large-distance structure of the theory, and so since in this phase the q, can easily vary by one unit as we move to neighboring lattice sites we expect that important configurations in, say, (2.2) will include those in which the q,.'s differ by arbitrarily large amounts over sufficiently long distances. Thus, for the purposes of determining the large distance structure, we should be able to replace the discrete sum over the qi's by a continuous integral over qi from - oo to oo*.

Of course for any non-zero fl there will be corrections to this approximation. These can be computed by using the identity

= foo d q ei2*rqk (2.10) q = - - o ¢ k = - o o - o o

and keeping successively higher values of I kl in the sum on the right hand side. But for fl < fl~ these corrections will affect only the short-distance structure of the theory; the long-distance behavior will be determined by the k = 0 term**.

With this in mind, we approximate (2.2) for fl < fll as

Z " ~ , / Dqexp N 2 X ( A t ~ q i + N l v ; , ) 2 , {/) oo ( )

fl < fl l ' ( 2 . 1 1 )

* Another way to say this is to imagine performing a real-space renormalization group procedure in which we group together spins in successively larger blocks. Since the nearest neighbor spins can freely vary by one unit, we expect that large enough blocks will include arbitrarily large values of the block spin variable with reasonably high probability. Thus, in this phase it should be a good large distance approximation to replace the discrete qi's by continuous variables.

** The terms with k=~ 0 correspond, in the language of Kadanoff (ref. [ l l D to electric vortices.

26 M . B . Einhorn et al. / Phase transitions

where the prime on the sum over ( l ) means a sum over (/} that produces distinct configurations of e~A~l~; i as discussed in subsect. 2.2. We can now do the gaussian

integral over the qi to obtain

Z = Z° Z(m} exp [ 2~r2fl ~ miV~jmj- cm2] (2.12)

The sum over rni is a sum over integer valued variables m i = e ~ A l , ; i on the sites of the dual lattice. From the discussion following (2.5) these are also the di- vergences of the topological current, i.e., m i = ( 1 / N ) A j v ; i. The potential V/jet I n [ i - j [ for [ i - j [ ~ > l, and c is a positive constant. The factor Z 0 is just the

gaussian integral in (2.11) with all l , = 0. Eqs. (2.1 l) and (2.12) are different forms of the periodic gaussian approximation

to the two-dimensional x-y model [12]. We can now easily understand the long-

distance structure of the system for fl < ill. First, we note that as a result of the domain boundaries becoming floppy for fl < ill, the intervortex potential has been softened from linear to logarithmic. This effect is easy to understand in terms of

our previous discussion of the x-y model. Remember that in the x-y model the vortex potential is logarithmic because the x-y model spins are continuous and can differ in orientation infinitesimally f rom their neighbors. The Z~v spins on the other hand can take on only a finite number of orientations. However, in the phase in which the Zu spin domain boundaries are floppy, one can imagine performing a thermal average over all possible configurations of domain boundaries, and so

generating infinitesimal relative rotations of nearest neighbor spins in an average sense. In the language of the renormalization group, we imagine making block spins as we scale to larger and larger distances. So long as all relative Z~v spin orientations occur with reasonable probability (as we expect for fl < fl~), then the block spin variable should take on an increasing number of possible orientations as the size of the block is increased, generating finally an effectively U(1) symmetric spin. [Note that this argument is not valid for the Ising case where the block spin variable is expected to become a one component (real) continuous field rather than

a two component (complex) field.] Finally, we note that, using (2.11), we can

compute (sisj). For fl > fll w e argued that this would approach a non-zero constant as r -- ]i - j [ - ~ o¢. But for fl < fll we expect that this will approach zero as r - , oe. That is does so is well-known from previous studies of the x-y model. The rate at which the correlation function decays for large r (whether algebraically or exponentially) will depend on fl and on the x-y model-like phase in which we find ourselves.

The phase in which we find ourselves depends on the behavior of the vortices. Since for fl < fll the vor tex-vor tex potential is logarithmic rather than linear, the vortices can now dissociate to form a plasma in a manner familiar f rom the x-y model. Following Kosterlitz and Thouless [13] one can easily approximate this dissociation temperature by writing down the free energy for a vor tex-ant ivor tex pair and noting the temperature above which the entropy dominates over the

M.B. Einhorn et al. / Phase transitions 27

energy. More sophisticated calculations of this dissociation temperature have also been done [16]. In terms of our variable r , the best value seems to be f l - - r 2

0.741. Now, if/32 < ill, then our theory will have three phases; a low-temperature ferromagnetic phase with small closed domain boundaries and vortex-ant ivortex pairs tightly bound by a linear potential, an intermediate phase in which the domain boundaries have condensed, but the vortices are still bound (albeit only logarithmically) and finally a completely disordered high-temperature phase in which the vortices unbind and condense to form a plasma of free charges. In the

middle phase, ]~2 < 1~ < /~l' G ( r ) = ( s i s j ) ~ r -~, and this phase thus has a kind of massless excitation, while for fl < f12, G( r ) ~ e - " r , indicating that in this phase a mass has been generated. If fll </32, then the massless phase will be absent: by the time the vortex potential has changed from linear to logarithmic due to the strings becoming floppy, the vortices already have sufficient entropy to unbind and so they condense at the same temperature as the domain boundaries. From (2.7) we see that fll depends on N, while fiE does not. Writing

/3 l = N 2 f N 2_1n3 29 2 2~ -2 '

we see that we should have three phases for N > N c -- 20r(f l2/2f )l/2,~v 3.65, while for N < N¢ we expect only two phases.

These results are in qualitative agreement with the work of refs. [8-10], although our crude estimate for fll incorrectly suggests two phase transitions for N -- 4 which is clearly wrong. But more important for our picture is that the dependence of fll on N is the same as that deduced in refs. [8-10], (see also subsect. 2.1) but from different arguments. Finally, we remark that since the model (2.2) is self-dual, the two phase transitions (assuming there are only two) must satisfy ri f t2 = N2/4°r2- Using our rough approximation for fll gives instead

a n ,,~ N2 / 21n3 ] N 2 - - 1 .63

p i p 2 - - 4~r2 k ~ ] - - ~ 4~r2

which is another measure of the crudeness of our approximation for fit. (A more accurate estimate of Arc may be obtained by enforcing duality. Assuming/32 ~ flxy ~0.741, then fl~ = N2/4~r2f12. But we have fit > r2 only for N > 2~rflxy ~4.66.)

We close this section by noting that since (2.2) is self-dual our entire picture can be reversed and we could discuss the phase transitions as condensations of electric vortices (replacing the magnetic strings) and electric domain boundaries (replacing the magnetic vortices). We have emphasized the totally magnetic picture, because given the original form of the theory, these variables are the most natural ones*.

* It is important to note that this same change from magnetic strings and vortices to electric vortices and strings is also possible in the vector Potts model even though that model is not self-duaL The

• ~e of self-duality means that the interactions among the electric variables will not be the same interactions among the magnetic variables. See also ref. [7].

28 M . B . Einhorn et al. / Phase transitions

3. Comments

3.1. Z N LATFICE G A U G E THEORIES A N D Z N SPIN MODELS IN HI GHER DIMENSIONS

Following our analysis of the two-dimensional Z N spin model, we should be able to easily describe the topological excitations of Z N spin models in higher di- mensions, and of Z N lattice gauge models, and perhaps say something about iheir phase properties. It should be clear that in d dimensions the topological excitation of the vector Potts model will be closed d - 1 dimensional domain boundaries (the analogue of our closed strings) and N open d - 1 dimensional manifolds which co-terminate on a d - 2 dimensional (closed) manifold (the analogue of our vortex-antivortex pair connected by N strings). The Zt¢ lattice gauge theory in d + 1 dimensions has the same topological excitations as the Z N vector Potts model in d dimensions. This follows the general pattern expected from the study of duality for abelian theories [7], and is simply related to the fact that the gauge theory interactions are defined on a 2-dimensional surface (elementary lattice plaquettes), while the spin theory has interactions defined on lattice links which are one- dimensional. Thus, in a fixed number of space dimensions, the dimension of the domain boundaries and their divergences (the "vortex'-l ike excitations) are re- duced by one for the gauge theory.

Using the fact that the periodic gaussian form of the four-dimensional Z N lattice gauge theory is self-dual, the authors of refs. [8-10] were able to generalize the arguments used for the two-dimensional Z N periodic gaussian spin model to show that for N > N¢ there must also be at least two phase transitions for the four- dimensional Z N lattice gauge theory*. From our point of view, the first (lower temperature) phase transition should correspond to a condensation of the closed two dimensional domain boundaries** taking us from a ferromagnetic to a massless phase, while the second phase transition should correspond to the con- densation of the open bounded manifolds. Unfortunately, it is not easy to make quantitative arguments in this case, because of the difficulty of computing the entropy of a two-dimensional sheet. Although not much is known about random two-dimensional flopping, qualitative arguments have been presented [18] which make this picture quite plausible.

Another case of interest is the 3-dimensional Z N spin model, which by duality can be transformed into the three-dimensional Z N lattice gauge theory. Again we have two types of excitations with topological significance. From the spin model point of view they are closed two-dimensional surfaces and open, bounded two- dimensional surfaces, while from the gauge model point of view they are closed

* The beautiful Monte Carlo studies of the four-dimensional Z N lattice gauge theories of ref. [17] a r e

a strong confirmation of the picture of two phase transitions for N > 5. ** Note that these are not really domain boundaries in the sense that they do not enclose a region of

four-space. They only enclose a region of three-space orthogonal to the direction defined by an excited vector potential. They are also gauge invariant.

M . B . Einhorn et al. / Phase transitions 29

strings and strings with monopoles on the ends. The latter picture is easier to analyze, and using it one would conclude that for general N there is only one phase transition. In the low-temperature phase the string free energy has its minimum at L -- string length = 0, and so there is a linear binding potential between monopole- antimonopole pairs. At the phase transition, when the strings become floppy, the monopole-antimonopole potential is softened to cr l / r , and so the monopole- antimonopole pairs unbind at the same temperature.

3.2. RELATION TO THE ELECTRIC AND MAGNETIC COULOMB GAS REPRESENTATION

Kandanoff and others [8-11] have shown that the two-dimensional Z N spin system has a representation in terms of a gas of interacting electric and magnetic charges. Similarly, the four-dimensional lattice gauge theory has a representation in terms of a soup of interacting electric and magnetic current loops [8-10]. The relationship between this representation and our picture is quite straightforward.

To help us understand this relationship, consider the periodic gaussian ap- proximation to the two-dimensional x-y model. This model has topological excita- tions which are vortex points interacting through a logarithmic potential. On the other hand, this model is dual to the two-dimensional discrete gaussian model. The "topological excitations" of the discrete gaussian model are the closed domain boundaries (strings) between islands of different values of the discrete gaussian field. Thus we see that in two dimensions we can replace a set of vortices by a set of strings. This is precisely analogous to the way in which the representation of Kadanoff differs from our picture. By undoing one of the duality transformations necessary to get the Coulomb gas representation, we are able to replace a set of electric charges by a set of magnetic strings. Of course both representations retain the magnetic charges which become the x-y model vortices in the limit N--~ 0¢. In a similar way, we can replace the electric current loops of the four-dimensional Z N gauge theory by magnetic sheets to get our representation in terms of magnetic sheets and magnetic sheets terminating on magnetic loops rather than electric loops and magnetic loops.

3.3. RELATION TO TWO-DIMENSIONAL MELTING

Recently Halperin and Nelson [5] have suggested that the melting of a two- dimensional crystal occurs via two separate phase transitions. Each phase transi- tion can be understood as being caused by the condensation of a different kind of topological defect of the crystal. Roughly, their picture can be described in the following way. First, they identify two kinds of defects, dislocations and disclina- tions. At low temperatures the dislocations are bound in pairs by a logarithmic potential, while the disclinations are bound by a potential which grows like the square of the separation. Next, they define separate correlation functions, F T and

30 M . B . Einhorn et al. / Phase transitions

F a which measure the degree of translational and rotational symmetry. For the triangular (square) lattice which they consider F R has the same form as the

sp in-spin correlation function for our d = 2 spin theory with a Z 6 (Z4) symmetry. In the low temperature phase of the crystal, F T ~ r - P and FR-->const ( ~ 0) as r---> or. Thus we have discretely broken rotational symmetry and a translational symmetry which would be broken in the usual way except that we are in two dimensions. As we raise the temperature, the dislocations suddenly unbind and condense into a plasma phase. In this middle phase F T ~ e - ~ " and F R ~ r - ~ as r--~ oo. Thus we have restored translational symmetry, but we still have long-range rotational order, although not of the ferromagnetic type (again, because we are in two dimensions). In addition, the plasma of dislocations screens the potential between the disclinations making it logarithmic and setting the stage for the next phase transition which occurs when the disclinations unbind. In the third, high- temperature phase, F T continues to fall exponentially at large distances and also F R ~ e -m" so we have a complete restoration of both translational and rotational

symmetry. In our model we have no simple analogue of the translational degrees of freedom

of the crystal, but our Z jr spins are clearly analogous to the crystal 's rotational

degrees of freedom. Thus both systems have a Z jr symmetry ( N - - 4 for a square crystal and N = 6 for a triangular crystal). In both systems there are two phase transitions (which are thought to be essential singularities), and the large-distance behaviors of F R in the crystal and of the spin-spin correlation function in our Z N models are the same in each of the three phases. Furthermore, in both systems each phase transition is caused by the unbinding and condensation of one of two types of topological excitations. Finally, in both systems the unbinding of the topological

excitations that gives rise to the second (high-temperature) phase transition is only possible because their binding energy is screened by the plasma of topological excitations whose condensation caused the first (low-temperature) phase transition.

Of course, the two systems are not identical. In particular, the energetics of analogous configurations in the two systems are not the same. This gives rise to

several differences, one of which is that even in the case of a square crystal with a Z 4 symmetry one expects two separate phase transitions for melting while for the Z 4 vector Potts or Villain model there is only one. However, because the underly- ing rotational symmetry of the two theories is the same their phase behavior is

quite analogous.

One of us (R.S.) would like to thank T. Banks, A. Schwimmer, T. Witten, and especially R. Pearson for very stimulating discussion. R.S. is also grateful to the Nuclear Physics Depar tment of the Weizmann Institute of Science and the Theory Division of CERN for their hospitality while part of this work was being done. E.R. appreciates the support of Lawrence Berkeley Laboratory, where a portion of

this work was performed.

M.B. Einhorn et al. / Phase transitions 31

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